*> \brief \b SLAIC1 applies one step of incremental condition estimation. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLAIC1 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C ) * * .. Scalar Arguments .. * INTEGER J, JOB * REAL C, GAMMA, S, SEST, SESTPR * .. * .. Array Arguments .. * REAL W( J ), X( J ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAIC1 applies one step of incremental condition estimation in *> its simplest version: *> *> Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j *> lower triangular matrix L, such that *> twonorm(L*x) = sest *> Then SLAIC1 computes sestpr, s, c such that *> the vector *> [ s*x ] *> xhat = [ c ] *> is an approximate singular vector of *> [ L 0 ] *> Lhat = [ w**T gamma ] *> in the sense that *> twonorm(Lhat*xhat) = sestpr. *> *> Depending on JOB, an estimate for the largest or smallest singular *> value is computed. *> *> Note that [s c]**T and sestpr**2 is an eigenpair of the system *> *> diag(sest*sest, 0) + [alpha gamma] * [ alpha ] *> [ gamma ] *> *> where alpha = x**T*w. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOB *> \verbatim *> JOB is INTEGER *> = 1: an estimate for the largest singular value is computed. *> = 2: an estimate for the smallest singular value is computed. *> \endverbatim *> *> \param[in] J *> \verbatim *> J is INTEGER *> Length of X and W *> \endverbatim *> *> \param[in] X *> \verbatim *> X is REAL array, dimension (J) *> The j-vector x. *> \endverbatim *> *> \param[in] SEST *> \verbatim *> SEST is REAL *> Estimated singular value of j by j matrix L *> \endverbatim *> *> \param[in] W *> \verbatim *> W is REAL array, dimension (J) *> The j-vector w. *> \endverbatim *> *> \param[in] GAMMA *> \verbatim *> GAMMA is REAL *> The diagonal element gamma. *> \endverbatim *> *> \param[out] SESTPR *> \verbatim *> SESTPR is REAL *> Estimated singular value of (j+1) by (j+1) matrix Lhat. *> \endverbatim *> *> \param[out] S *> \verbatim *> S is REAL *> Sine needed in forming xhat. *> \endverbatim *> *> \param[out] C *> \verbatim *> C is REAL *> Cosine needed in forming xhat. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realOTHERauxiliary * * ===================================================================== SUBROUTINE SLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C ) * * -- LAPACK auxiliary routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER J, JOB REAL C, GAMMA, S, SEST, SESTPR * .. * .. Array Arguments .. REAL W( J ), X( J ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 ) REAL HALF, FOUR PARAMETER ( HALF = 0.5E0, FOUR = 4.0E0 ) * .. * .. Local Scalars .. REAL ABSALP, ABSEST, ABSGAM, ALPHA, B, COSINE, EPS, \$ NORMA, S1, S2, SINE, T, TEST, TMP, ZETA1, ZETA2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, SIGN, SQRT * .. * .. External Functions .. REAL SDOT, SLAMCH EXTERNAL SDOT, SLAMCH * .. * .. Executable Statements .. * EPS = SLAMCH( 'Epsilon' ) ALPHA = SDOT( J, X, 1, W, 1 ) * ABSALP = ABS( ALPHA ) ABSGAM = ABS( GAMMA ) ABSEST = ABS( SEST ) * IF( JOB.EQ.1 ) THEN * * Estimating largest singular value * * special cases * IF( SEST.EQ.ZERO ) THEN S1 = MAX( ABSGAM, ABSALP ) IF( S1.EQ.ZERO ) THEN S = ZERO C = ONE SESTPR = ZERO ELSE S = ALPHA / S1 C = GAMMA / S1 TMP = SQRT( S*S+C*C ) S = S / TMP C = C / TMP SESTPR = S1*TMP END IF RETURN ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN S = ONE C = ZERO TMP = MAX( ABSEST, ABSALP ) S1 = ABSEST / TMP S2 = ABSALP / TMP SESTPR = TMP*SQRT( S1*S1+S2*S2 ) RETURN ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN S1 = ABSGAM S2 = ABSEST IF( S1.LE.S2 ) THEN S = ONE C = ZERO SESTPR = S2 ELSE S = ZERO C = ONE SESTPR = S1 END IF RETURN ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN S1 = ABSGAM S2 = ABSALP IF( S1.LE.S2 ) THEN TMP = S1 / S2 S = SQRT( ONE+TMP*TMP ) SESTPR = S2*S C = ( GAMMA / S2 ) / S S = SIGN( ONE, ALPHA ) / S ELSE TMP = S2 / S1 C = SQRT( ONE+TMP*TMP ) SESTPR = S1*C S = ( ALPHA / S1 ) / C C = SIGN( ONE, GAMMA ) / C END IF RETURN ELSE * * normal case * ZETA1 = ALPHA / ABSEST ZETA2 = GAMMA / ABSEST * B = ( ONE-ZETA1*ZETA1-ZETA2*ZETA2 )*HALF C = ZETA1*ZETA1 IF( B.GT.ZERO ) THEN T = C / ( B+SQRT( B*B+C ) ) ELSE T = SQRT( B*B+C ) - B END IF * SINE = -ZETA1 / T COSINE = -ZETA2 / ( ONE+T ) TMP = SQRT( SINE*SINE+COSINE*COSINE ) S = SINE / TMP C = COSINE / TMP SESTPR = SQRT( T+ONE )*ABSEST RETURN END IF * ELSE IF( JOB.EQ.2 ) THEN * * Estimating smallest singular value * * special cases * IF( SEST.EQ.ZERO ) THEN SESTPR = ZERO IF( MAX( ABSGAM, ABSALP ).EQ.ZERO ) THEN SINE = ONE COSINE = ZERO ELSE SINE = -GAMMA COSINE = ALPHA END IF S1 = MAX( ABS( SINE ), ABS( COSINE ) ) S = SINE / S1 C = COSINE / S1 TMP = SQRT( S*S+C*C ) S = S / TMP C = C / TMP RETURN ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN S = ZERO C = ONE SESTPR = ABSGAM RETURN ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN S1 = ABSGAM S2 = ABSEST IF( S1.LE.S2 ) THEN S = ZERO C = ONE SESTPR = S1 ELSE S = ONE C = ZERO SESTPR = S2 END IF RETURN ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN S1 = ABSGAM S2 = ABSALP IF( S1.LE.S2 ) THEN TMP = S1 / S2 C = SQRT( ONE+TMP*TMP ) SESTPR = ABSEST*( TMP / C ) S = -( GAMMA / S2 ) / C C = SIGN( ONE, ALPHA ) / C ELSE TMP = S2 / S1 S = SQRT( ONE+TMP*TMP ) SESTPR = ABSEST / S C = ( ALPHA / S1 ) / S S = -SIGN( ONE, GAMMA ) / S END IF RETURN ELSE * * normal case * ZETA1 = ALPHA / ABSEST ZETA2 = GAMMA / ABSEST * NORMA = MAX( ONE+ZETA1*ZETA1+ABS( ZETA1*ZETA2 ), \$ ABS( ZETA1*ZETA2 )+ZETA2*ZETA2 ) * * See if root is closer to zero or to ONE * TEST = ONE + TWO*( ZETA1-ZETA2 )*( ZETA1+ZETA2 ) IF( TEST.GE.ZERO ) THEN * * root is close to zero, compute directly * B = ( ZETA1*ZETA1+ZETA2*ZETA2+ONE )*HALF C = ZETA2*ZETA2 T = C / ( B+SQRT( ABS( B*B-C ) ) ) SINE = ZETA1 / ( ONE-T ) COSINE = -ZETA2 / T SESTPR = SQRT( T+FOUR*EPS*EPS*NORMA )*ABSEST ELSE * * root is closer to ONE, shift by that amount * B = ( ZETA2*ZETA2+ZETA1*ZETA1-ONE )*HALF C = ZETA1*ZETA1 IF( B.GE.ZERO ) THEN T = -C / ( B+SQRT( B*B+C ) ) ELSE T = B - SQRT( B*B+C ) END IF SINE = -ZETA1 / T COSINE = -ZETA2 / ( ONE+T ) SESTPR = SQRT( ONE+T+FOUR*EPS*EPS*NORMA )*ABSEST END IF TMP = SQRT( SINE*SINE+COSINE*COSINE ) S = SINE / TMP C = COSINE / TMP RETURN * END IF END IF RETURN * * End of SLAIC1 * END